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On an analogue of BRK-type sets in finite fields
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 13 July 2026, pp. 1-14
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A Besicovitch–Rado–Kinney (BRK) set in $\mathbb {R}^n$
contains a hypersphere of every radius. In $\mathbb {F}_q^n$
, BRK-type sets of degree $\ell $
analogously contain a family of $(n-1)$
-dimensional surfaces, parameterized by a dilation factor and determined by a fixed homogeneous polynomial of degree $\ell $
. We define $(n,d)$
-BRK-type sets of degree $\ell $
, which contain a family of d-dimensional sets parameterized by an $(n-d)$
-dimensional dilation factor and determined by fixed homogeneous polynomials of degree $\ell $
. We use the polynomial method to obtain a lower bound $|S| \gtrsim _{n, \ell } q^n$
on $(n,d)$
-BRK-type sets S of degree $\ell $
. We obtain an improved lower bound $|S| \geq \frac {(q-1)^n}{(\ell + 1 - 2\ell /q)^n}$
by implementing the method of multiplicities; this is the same bound obtained by Trainor on BRK-type sets of degree $\ell $
, and we obtain this bound independently of d.
